Hello all,

Haven't posted here in awhile...

I posted this message, Venn diagram method of seeing intersections in n-space... bivectors in 4-space... on the Math Message Board.

Unfortunately, few there show much interest in higher-dimensional geometry. No one has responded to my post.

I've answered some of my questions myself, but I'm still confused about what precisely Lounestro might have meant concerning that decomposition of a non-simple 4D bivector...?

There's just something odd about his statement... and I'm just unsure precisely what he meant. For one thing, does one need to perform the decomposition on the right-hand side in order to have two orthogonal bivectors? Aren't the two bivectors on the left-hand side also orthogonal?

And, also... as I mention in one of my follow on responses... I'm also pretty sure Lounestro is using the convention that a bivector doesn't refer to a specific parallogram... but rather, that all we keep is its directed magnitude, so we only know that the bivector represents some directed area in the 2D-plane spanned by its component 1-vectors.

Anyway... the long and the short of it is that I don't Lounestro's assertion is true for general 2D-planes in 4-space, which utilizing the convention I wish to use make reference to a specific parallogram by keeping the information on the component 1-vectors and defining that area as the specific area defined by the bivector.

Further, to define arbitrary 1-vectors one needs two position vectors... or else all 1-vectors are limited to going through the origin. That is, there's no distinction between a 1-vector and all it's translates.

One way or another, I don't believe Lounestro's assertion could hold with bivectors describing planes not going through the origin...

Does anyone have ideas about what precisely Lounestro meant? And/or any comments, or observations, on my Venn diagram method of seeing these intersections in n-space...?